Page 333 - Elements of Distribution Theory

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10.5 Exercises 319

we may use Gaussian quadrature based on the Laguerre polynomials. For illustration,
consider the case n = 5. The zeros of L 5 and the corresponding weights λ 0 ,...,λ 5 are
available, for example, in Abramowitz and Stegun (1964, Table 25.9). Using these values,
an approximation to E[g(X)] is, roughly,

.5218g(.26) + .3987g(1.41) + .0759g(3.60) + .0036g(7.09) + .00002g(12.64);
for the numerical calculations described below, 10 significant figures were used.
Recall that this approximation is exact whenever g is a polynomial of degree 9 or less. In
general, the accuracy of the approximation will depend on how well g may be approximated
by a 9-degree polynomial over the interval (0, ∞). For instance, suppose g(x) = sin(x);
then E[g(X)] = 1/2 while the approximation described above is 0.49890, a relative error
1 √
of roughly 0.2%. However, if g(x) = x − 2 , then E[g(X] = π/2 and the approximation is
1.39305, an error of roughly 21%.

10.5 Exercises

10.1 Prove Lemma 10.1.
10.2 Let F denote the distribution function of the absolutely continuous distribution density function

2
2x exp{−x }, x > 0.
Find the first three orthogonal polynomials with respect to F.
10.3 Let F denote the distribution function of the discrete distribution with frequency function
1
, x = 0, 1,....
2 x+1
Find the first three orthogonal polynomials with respect to F.
10.4 Let F denote a distribution function on R such that all moments of the distribution exist and the
distribution is symmetric about 0, in the sense that
F(x) + F(−x) = 1, −∞ < x < ∞.

Let p 0 , p 1 ,... denote orthogonal polynomials with respect to F. Show that the orthogonal
polynomials of even order include only even powers of x and that the orthogonal polynomials
of odd order include only odd powers of x.
10.5 Let {p 0 , p 1 ,...} denote orthogonal polynomials with respect to a distribution function F and
suppose that the polynomials are standardized so that

∞
2
p j (x) dF(x) = 1, j = 0, 1,....
−∞
Fix n = 0, 1,... and define a function K n : R × R → R by
K n (x, y) = n
p j (x)p j (y).
j=0
Show that, for any polynomial q of degree n or less,
∞

q(x) = K n (x, y)q(y) dF(y).
−∞